This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A074816 #47 Dec 27 2024 04:11:35
%S A074816 1,3,3,3,3,9,3,3,3,9,3,9,3,9,9,3,3,9,3,9,9,9,3,9,3,9,3,9,3,27,3,3,9,9,
%T A074816 9,9,3,9,9,9,3,27,3,9,9,9,3,9,3,9,9,9,3,9,9,9,9,9,3,27,3,9,9,3,9,27,3,
%U A074816 9,9,27,3,9,3,9,9,9,9,27,3,9,3,9,3,27,9,9,9,9,3,27,9,9,9,9,9,9,3,9,9,9
%N A074816 a(n) = 3^A001221(n) = 3^omega(n).
%C A074816 Old name was: a(n) = sum(d|n, tau(d)*mu(d)^2 ).
%C A074816 Terms are powers of 3.
%C A074816 The inverse Mobius transform of A074823, as the Dirichlet g.f. is product_{primes p} (1+2*p^(-s)) and the Dirichlet g.f. of A074816 is product_{primes p} (1+2*p^(-s))/(1-p^(-s)). - _R. J. Mathar_, Feb 09 2011
%C A074816 If n is squarefree, then a(n) = #{(x, y) : x, y positive integers, lcm (x, y) = n}. See Crandall & Pomerance. - _Michel Marcus_, Mar 23 2016
%D A074816 Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 2.3 p. 108.
%H A074816 R. Zumkeller, <a href="/A074816/b074816.txt">Table of n, a(n) for n = 1..10000</a>
%F A074816 a(n) = 3^m if n is divisible by m distinct primes. i.e., a(n)=3 if n is in A000961; a(n)=9 if n is in A007774 ...
%F A074816 a(n) = 3^A001221(n) = 3^omega(n). Multiplicative with a(p^e)=3. - _Vladeta Jovovic_, Sep 09 2002.
%F A074816 a(n) = tau_3(rad(n)) = A007425(A007947(n)). - _Enrique Pérez Herrero_, Jun 24 2010
%F A074816 a(n) = abs(Sum_{d|n} A000005(d^3)*mu(d)). - _Enrique Pérez Herrero_, Jun 28 2010
%F A074816 a(n) = Sum_{d|n, gcd(d, n/d) = 1} 2^omega(d) (The total number of unitary divisors of the unitary divisors of n). - _Amiram Eldar_, May 29 2020, Dec 27 2024
%F A074816 a(n) = Sum_{d1|n, d2|n} mu(d1*d2)^2. - _Wesley Ivan Hurt_, Feb 04 2022
%F A074816 Dirichlet g.f.: zeta(s)^3 * Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)). - _Vaclav Kotesovec_, Feb 16 2022
%t A074816 A074816[n_]:=3^PrimeNu[n]; (* _Enrique Pérez Herrero_, Jun 28 2010 *)
%o A074816 (PARI) a(n) = 3^omega(n); \\ _Michel Marcus_, Mar 23 2016
%o A074816 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X)/(1 - X))[n], ", ")) \\ _Vaclav Kotesovec_, Feb 16 2022
%Y A074816 Cf. A001221, A034444, A069212, A124508.
%K A074816 nonn,mult
%O A074816 1,2
%A A074816 _Benoit Cloitre_, Sep 08 2002
%E A074816 Simpler definition at the suggestion of _Michel Marcus_. - _N. J. A. Sloane_, Mar 25 2016